A290225 Number of solid standard Young tableaux of cylindrical shape lambda X n, where lambda ranges over all partitions of n.

1, 1, 4, 276, 1161678, 620383261034, 80434777704834144228, 3212151962391797592956111856142, 58141033434729590882944205957642581926272684, 738506234630963222745737660670442498620046849638365979249010

Offset: 0

Author

Alois P. Heinz, Jul 24 2017

Keywords

• nonn

Links

• S. B. Ekhad and D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229 [math.CO], 2012.
• Wikipedia, Young tableau

Crossrefs

Main diagonal of A215204.

Programs

Maple

b:= proc(l) option remember; local m; m:= nops(l);
      `if`({map(x-> x[], l)[]}minus{0}={}, 1, add(add(`if`(l[i][j]>
      `if`(i=m or nops(l[i+1])
      `if`(nops(l[i])=j, 0, l[i][j+1]), b(subsop(i=subsop(
       j=l[i][j]-1, l[i]), l)), 0), j=1..nops(l[i])), i=1..m))
    end:
g:= proc(n, i, k, l) `if`(n=0 or i=1, b(map(x-> [k$x], [l[], 1$n])),
       add(g(n-i*j, i-1, k, [l[], i$j]), j=0..n/i))
    end:
a:= n-> g(n$3, []):
seq(a(n), n=0..6);

Mathematica

b[l_] := b[l] = With[{m = Length[l]}, If[Union[l // Flatten] ~Complement~ {0} == {}, 1, Sum[Sum[If[l[[i, j]] > If[i == m || Length[l[[i + 1]]] < j, 0, l[[i + 1, j]]] && l[[i, j]] > If[Length[l[[i]]] == j, 0, l[[i, j + 1]]], b[ReplacePart[l, i -> ReplacePart[l[[i]], j -> l[[i, j]] - 1]]], 0], {j, 1, Length[l[[i]]]}], {i, 1, m}]]];
g[n_, i_, k_, l_] := If[n == 0 || i == 1, b[Table[k, {#}] & /@ Join[l, Table[1, {n}]]], Sum[g[n - i*j, i - 1, k, Join[l, Table[i, {j}]]], {j, 0, n/i}]];
a[n_] := g[n, n, n, {}];
Table[a[n], {n, 0, 7}] (* Jean-François Alcover, Dec 28 2022, after Alois P. Heinz in A215204 *)

Formula

a(n) = A215204(n,n).